I've been speculatively looking at the P2 V70/XC70. My potential price range is right around yours. I have heard that the 01-02 transmissions have real problems, however the cars I've seen all have, give or take, about 100-125K miles on them and look to be in very good shape. It seems to be that most people fear the Tiptronic transmissions (unless I'm wrong) that come in the T5 V70 and standard on the XC70. I'm not exactly sure what even goes wrong on these transmissions or what signs to look for.
What happened to the 01 & 02 model transmissions was they were equipped with Volvo's stop neutral software when new. This basically caused the car to shift into neutral when say stopping at a red light. I think the intention was to lower emissions and save fuel. The transmission itself were/are the same Aisin Warner transmissions used in other Volvo's and other makes as well with no problem (minus the stop neutral feature). The stop neutral function if not removed through software by the dealer at some point during the tranny's life (preferably early) has and can cause internal damage to the transmission. Clunking, hard shifting, slipping, and overall failure are symptoms of this condition.
I know there are probably many people who have these model cars and like them very much and there are people who have tons of miles on them and may not have done anything to their cars service wise. But, any forum on the web related to these cars has tons and tons of threads regarding early P2 model S60's and V70's with automatic transmission problems.
The next issue is the cost of repair/replacement. If you buy a used transmission from a boneyard you take a chance not getting something any better than you had (newer models aren't compatible) and rebuilds reportedly run in the 3500-5k range if you can even get anyone competent to do it. The early P2 cars also tend to have more annoyance type issues (ie; electrical) than some of the later models. I think this can be attributed to the cars now having some age and the typical work out the bugs type issues many manufacturers have when a new model or platform is introduced.
This in a nutshell is why many here and on various other forums and enthusiast sites will recommend to stay away from these models.